本文關(guān)鍵詞： 有限性2-范疇 Duflo對合 箭圖 單可遷2-表示 Drinfeld中心　出處：《華東師范大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：本文刻畫了有限性2-范疇的抽象Duflo對合(由每個給定的左胞腔唯一確定).同時研究了一些與(代數(shù)閉域上)有限維代數(shù)相關(guān)的有限性2-范疇的結(jié)構(gòu)和表示.首先.在fiat2-范疇中.通過左胞腔對應(yīng)的Duflo對合可以定義阿貝爾胞腔2-表示,這是2-版本意義上的一類不可約表示.在有限性2-范疇中,通過其主2-表示的子商形式可以定義加性胞腔2-表示.而且.在fiat情形下,對于任意的左胞腔,如此得到的加性胞腔2-表示的阿貝爾化與通過Duflo對合得到的阿貝爾胞腔2-表示是互等價的.受到flat2-范疇中的Dufl。對合定義的啟發(fā).我們在任意的有限性2-范疇中給出了類似的定義,并且證明了對于文中三類有限性2-范疇的任意左胞腔如此定義的Dufl。對合都是存在的,其中兩類是與有限樹箭圖的路代數(shù)(簡稱樹路代數(shù))相關(guān)的：一類是由對偶投射函子確定的有限性2-范疇.另-類是由對偶投射函子和投射雙模共同確定的有限性2-范疇.顯然,后者包含前者作為其2-子范疇.不同十flat情形.有限性情形下的Duflo對合可能不落于所給左胞腔中.同時,我們描述了這兩類有限性2-范疇的主2-表示的底代數(shù)的箭圖,它們提供了相應(yīng)的阿貝爾主2-表示作用在對象上所得范疇的一些信息,即等價于相應(yīng)底代數(shù)的模范疇.其次,在有限性2-范疇中,單可遷2-表示可以看成“單”的2-表示.事實(shí)上對于任何有限性2-表示,都可構(gòu)造它的一個弱的Jordan-Holder列且其弱的合成子商都是單可遷2-表示,得到相應(yīng)的弱的Jordan-Holder定理.因此.具體的有限性2-范疇的單可遷2-表示的分類問題是非常有意義的.本文中,我們分類了上述與樹路代數(shù)相關(guān)的第一類有限性2-范疇上的所有單可遷2-表示.同時,我們也考慮了上述三類中由有限維代數(shù)的投射雙模確定的那類有限性2-范疇,我們所研究的是涉及的有限維代數(shù)非內(nèi)射的情形.但是我們目前無法給出一般的分類情況.然而在其中兩種較小情形下.我們分類了此類有限性2-范疇的所有單可遷2-表示.對于樹路代數(shù),我們定義了其上的可補(bǔ)理想,并構(gòu)造了一類新的有限性2-范疇,而且分類了A。型定向箭圖情形時的所有單可遷2-表示.對于這幾類有限性2-范疇,我們都有結(jié)論：每個單可遷2-表示都等價于一個胞腔2-表示.然而,對于復(fù)數(shù)域上截頭多項(xiàng)式代數(shù)的一類fiat2-范疇,此結(jié)論并不成立,它含有非胞腔2-表示的單可遷2-表示.最后,我們考慮了如何計算具體的有限性2-范疇的Drinfeld中心,它可以看成2-范疇中恒等2-函子的自同態(tài)范疇,是一個辮子monoidal范疇.在文中最后一部分,我們分別計算了上述樹路代數(shù)的對偶投射函子的有限性2-范疇,An型定向箭圖路代數(shù)可補(bǔ)理想的有限性2-范疇和截頭多項(xiàng)式代數(shù)的fiat2-范疇的Drinfeld中心,其中一類的Drinfeld中心雙等價于它的態(tài)射范疇,另一類的Drinfeld中心的不可分解對象是恒等1-態(tài)射確定的一些對.
[Abstract]:In this paper, we characterize the abstract Duflo involution of finiteness 2-category (determined by the uniqueness of each given left cell). At the same time, we study the structure and representation of some finiteness 2-categories related to finite dimensional algebras (over algebraic closed fields). In the fiat 2-category, the Abelian cell 2-representation can be defined by the Duflo involution corresponding to the left cell. This is a class of irreducible representations in the sense of 2-version. In the finiteness 2-category, the additive cell 2-representation can be defined by the subquotient form of its principal 2-representation. Moreover, in the case of fiat, for any left cell, The Abelization of the additive 2-representation and the Abelian 2-representation obtained by Duflo involution are mutually equivalent. It is inspired by the definition of Dufl-involution in the flat2-category. Have come up with a similar definition, And it is proved that the Dufl. involution of any left cell of the three finiteness 2-categories in this paper exists. Two of them are related to the path algebra of finite tree quiver (tree path algebra for short): one is finiteness 2-category determined by dual projective functor, and the other is finite determined by dual projective functor and projective bimodules. Sex 2-Category. Obviously, The latter includes the former as its 2-subcategory. Different ten flat cases. The Duflo involution in finiteness case may not fall into the given left cell. At the same time, we describe the quiver of the base algebra of the principal 2-representation of the two finiteness 2-categories. They provide some information about the category of the corresponding Abelian principal 2-representation action on the object, that is, it is equivalent to the module category of the corresponding bottom algebra. Secondly, in the finiteness 2-category, A simple transitive 2-representation can be regarded as a 2-representation of "simple". In fact, for any finite 2-representation, a weak Jordan-Holder column can be constructed and its weak compositons quotient is a simple transitive 2-representation. The corresponding weak Jordan-Holder theorem is obtained. Therefore, it is very meaningful to obtain the classification problem of the simple transitive 2-representation of the specific finiteness 2-category. In this paper, We classify all the simple transitive 2-representations of the first class of finiteness 2-category related to tree path algebras. At the same time, we also consider the class of finiteness 2-categories determined by projective bimodules of finite-dimensional algebras in the above three classes. What we are studying is the case of finite dimensional algebras that are not injective. However, we can not give a general classification at present. However, in two smaller cases, we classify all of the finiteness 2-categories. Transitive 2-representation. For tree path algebra, In this paper, we define complementary ideals on them, construct a new class of finiteness 2-categories, and classify all simple transitive 2-representations in the case of A. type directed quiver. We all have a conclusion that every simple transitive 2-representation is equivalent to a 2-representation in a cell. However, for a class of fiat2-category of truncated polynomial algebras over complex fields, this conclusion does not hold true. It contains a simple transitive 2-representation of non-cellular 2-representation. Finally, we consider how to calculate the Drinfeld center of a specific finiteness 2-category, which can be regarded as an endomorphism category of a constant iso-functor in a 2-category. Is a braided monoidal category. We calculate the finiteness of the dual projective functors of the tree path algebras mentioned above. The Drinfeld centers of the 2-category and the fiat2-category of the truncated polynomial algebras are calculated respectively. One kind of Drinfeld center is equivalent to its morphism category, the other kind of indecomposable object of Drinfeld center is some pairs of identity 1-morphism determinations.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O154.1

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